Defective number

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Demonstration with the Cuisenaire strips of the number 8 deficiency

In number theory, a defective number or deficient number is a number n for which the sum of its divisors is less than 2n. Equivalently, is a number for which the sum of its proper divisors (or aliquot sum) is less than n. For example, the proper divisors of 8 are 1, 2, and 4, and their sum is less than 8, so 8 is deficient.

Denoting by σ(n) the sum of the divisors, the value 2n − σ (n) is the deficiency of the number n. In terms of the aliquot sum s(n), the deficiency is n - s(n ).

Examples

The first poor numbers are

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50,... (A005100 Succession in OEIS)

As an example, consider the number 21. Its divisors are 1, 3, 7, and 21, and their sum is 32. Since 32 is less than 42, the number 21 is deficient. Its deficiency is 2 × 21 − 32 = 10.

Properties

Since the aliquot sums of the prime numbers are equal to 1, all prime numbers are deficient. More generally, all odd numbers with one or two distinct prime factors are deficient. It follows that there are infinitely many odd deficient numbers. There is also an infinite number of even deficient numbers, since all powers of two have the sum ( $1 + 2 + 4 + 8 +... + 2 x -1 = 2 x - 1$ ).

More generally, all premium powers ${displaystyle p^{k}}$ are deficient because their only divisors are ${displaystyle 1,p,p^{2},dotsp^{k-1}}$ That they add ${displaystyle {frac {p^{k}-1}{p-1}}}$ It's like a lot. ${displaystyle p^{k}-1}$ .

All proper divisors of deficient numbers are deficient. Furthermore, all proper divisors of perfect numbers are deficient.

There is at least one deficient number in the interval ${displaystyle [n,n+(log n)^{2}]}$ for everything n Big enough.

Related concepts

Euler's chart of numbers less than 100, classified as abundant, abundant primitives, abundant, superabundant, colossally abundant, highly composite, highly composite superior, strange and perfect in relation to its deficiency and composition of factors

Closely related to the deficient numbers are the perfect numbers with σ(n) = 2n, and the abundant numbers with σ(n) > 2n.

Natural numbers were first classified as deficient, perfect, or abundant by Nicomacheus of Gerasa in his Introductio Arithmetica (circa 100 AD).

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